Upcrossings of Random Fields

is of great importance in many applications. For example, if we consider a geographical map and denote height by X(t) where t is the set of geographical coordinates, Z(S) is the height of the highest mountain in the area S. In general, it is not possible to make any exact useful statements about the distribution of Z(S), and one must have recourse to approximations. One useful way of obtaining such approximations is to consider certain point processes in R" whose properties are related to those of the maximum. We shall consider in particular two such processes:

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[4]  R. Adler.  On the envelope of a Gaussian random field , 1978, Journal of Applied Probability.